In noncommutative geometry, the Jaffe- Lesniewski-Osterwalder (JLO) cocycle (named after Arthur Jaffe, Andrzej Lesniewski, and Konrad Osterwalder) is a cocycle in an entire cyclic cohomology group. It is a non-commutative version of the classic Chern character of the conventional differential geometry. In noncommutative geometry, the concept of a manifold is replaced by a noncommutative algebra \mathcal{A} of "functions" on the putative noncommutative space. The cyclic cohomology of the algebra \mathcal{A} contains the information about the topology of that noncommutative space, very much as the de Rham cohomology contains the information about the topology of a conventional manifold. The JLO cocycle is associated with a metric structure of non-commutative differential geometry known as a \theta-summable spectral triple (also known as a \theta-summable Fredholm module). It was first introduced in a 1988 paper by Jaffe, Lesniewski, and Osterwalder.

\theta -summable spectral triples

The input to the JLO construction is a \theta-summable spectral triple. These triples consists of the following data: (a) A Hilbert space \mathcal{H} such that \mathcal{A} acts on it as an algebra of bounded operators. (b) A -grading \gamma on \mathcal{H},. We assume that the algebra \mathcal{A} is even under the -grading, i.e., for all. (c) A self-adjoint (unbounded) operator D, called the Dirac operator such that A classic example of a \theta-summable spectral triple arises as follows. Let M be a compact spin manifold,, the algebra of smooth functions on M, \mathcal{H} the Hilbert space of square integrable forms on M, and D the standard Dirac operator.

The cocycle

Given a \theta-summable spectral triple, the JLO cocycle associated to the triple is a sequence of functionals on the algebra \mathcal{A}, where for n=2,4,\dots. The cohomology class defined by is independent of the value of t